Irreducible Harish-Chandra modules over extended Witt algebras

c 2012 by Institut Mittag-Leffler. All rights reserved

Irreducible Harish-Chandra modules over extended Witt algebras

Xiangqian Guo, Genqiang Liu and Kaiming Zhao

Abstract. Letdbe a positive integer,A=C[t^±₁¹, ..., t^±_d¹] be the Laurent polynomial algebra, and W=Der(A) be the derivation Lie algebra of A. Then we have the semidirect product Lie algebraWAwhich we call the extended Witt algebra of rankd. In this paper, we classify all irreducible Harish-Chandra modules overWAwith nontrivial action ofA.

1. Introduction

The Witt algebraW of rankdis the Lie algebra consisting of all derivations of the Laurent polynomial algebra A=C[t^±₁¹, t^±₂¹, ..., t^±_d¹], where dis a fixed positive integer. This algebra is isomorphic to the algebra of diffeomorphisms of the d- dimensional torus. The algebra W is a natural higher-rank generalization of the centerless Virasoro algebra, which has wide applications in different branches of mathematics and physics (see [9] and [11]–[15]). Unlike the Virasoro algebra, when d>1,W is centrally closed and its representation theory is still under development.

Modules over Witt algebras were used by O. Mathieu [18] to model simple cuspidal weight modules with finite-dimensional weight spaces over simple finite- dimensional Lie algebras. Representations of Witt algebras are also closely con- nected with the representation theory of extended affine Lie algebras ([1]) and toroidal Lie algebras ([2], [7], and [8]).

Representation theory of Witt algebras has been intensively studied by many mathematicians and physicists, see [3], [5], [6], [10], [11]–[15], [19], and [22]. In particular, [19] asserted recently that any irreducible Harish-ChandraW-module is either dense (with uniformly bounded weight spaces) or punctured (with uniformly bounded weight spaces) or a simple quotient of some generalized Verma module. So far, the only known dense or punctured modules are subquotients of the modules F^α(ψ, e) (see Section2.1) introduced and studied in [13] and [21], which are called

Shen modules or Larsson modules. Eswara Rao (see the introduction of [6]) said that it will be interesting to prove that any irreducible Harish-ChandraW-module is either a highest weight module or a Larsson module up to a twist of a GL(d,Z) action. This was formulated into a more precise conjecture in [10] as follows.

Conjecture. All nontrivial irreducible uniformly bounded modules over W are isomorphic to irreducible subquotients ofF^α(ψ, e).

Assuming this conjecture and using the result in [4], we can deduce that the third class of modules mentioned above depend only on the first two classes of the modules overW. So it is crucial to classify dense modules and punctured modules overW.

As an attempt to obtain a proof of this conjecture, we consider representations of the Lie algebraWA, which was started by Eswara Rao in [6]. TheW-modules F^α(ψ, e) can be naturally made into (WA)-modules. In [6], Eswara Rao proved that if the action of A is associative on an irreducible (WA)-module V, then V must be isomorphic to F^α(ψ, e). In Section 3, we show that the condition of Eswara Rao is always satisfied for any irreducible uniformly bounded (WA)- module with nontrivial action of A, up to a natural automorphism of WA. As a result, any nontrivial irreducible uniformly bounded (WA)-module is either an irreducible uniformly boundedW-module if the action ofA is zero, or isomorphic to some F^α(ψ, e, c) (a deformation of F^α(ψ, e), see Section 2 for its definition) if the action of A is nonzero. At last in Section 4, we show that any nontrivial irreducible (WA)-module which is not uniformly bounded must be isomorphic to the unique quotient module of some generalized Verma module induced from a nontrivial irreducible uniformly bounded module over some subalgebra isomorphic to Wd−1Ad−1, where Ad−1=C[t^±₁¹, ..., t^±_d−¹₁]. If the conjecture mentioned above holds, we can give a complete classification of irreducible Harish-Chandra modules overWA, using our results in Sections3and4.

2. Notation and preliminaries

We denote byZ,Z⁺,N,RandCthe sets of all integers, nonnegative integers, positive integers, real numbers and complex numbers, respectively. For any positive integer k, denote by Mk(C) the set of all k×k matrices over C. For a matrix B∈M_k(C), we denote its (i, j)-entry byB(i, j). For a Lie algebraL, we denote its enveloping algebra byU(L).

2.1. Witt algebraW and the extended Witt algebra WA

Throughout this paper, we fix a positive integer d>1. Let C^d be the vector space ofd×1 matrices with the standard basis{e1, e2, ..., ed} andZ^d be the subset of C^d with entries in Z. Let (· | ·) be the standard symmetric bilinear form such that (u|v)=u^Tv∈C, whereu^T is the matrix transpose of u.

LetA=C[t^±₁¹, ..., t^±_d¹] be theLaurent polynomial algebra overCand denote by W the algebra of all derivations ofA, called the Witt algebra.

Set tⁿ=tⁿ₁¹...tⁿ_d^d for n=(n1, n2, ..., nd)^T∈Z^d and ∂i=ti∂/∂ti for i∈{1, ..., d}. Set D(γ, m)=t^md

i=1γ_i∂_i for any γ∈C^d and m∈Z^d. Then W is spanned by all D(γ, m) withγ∈C^d andm∈Z^d. The Lie bracket ofW is given by

(1) [D(γ, m), D(δ, n)] =D(ε, m+n) forγ, δ∈C^d andm, n∈Z^d,

whereε=(γ|n)δ−(δ|m)γ. The extended Witt algebraWA=W⊕Ais a Lie alge- bra by extending the Lie structure ofW in the following way:

(2) [t^m, tⁿ] = 0 and [D(γ, m), tⁿ] = (γ|n)t^m+n for any γ∈C^d andm, n∈Z^d. Both W and WA are Z^d-graded, and their homogeneous subspaces are defined by (W)m=

γ∈C^dD(γ, m) and (WA)m=(W)m⊕Ct^mfor anym∈Z^d.

TakeL=WorL=WA. ThenL0is the Cartan subalgebra ofL. AnL-module V is called aweight moduleif the action ofL0onV is diagonalizable. For any weight moduleV we have the decompositionV=

λ∈L^∗0Vλ,where L^∗0=Hom_C(L0,C) and Vλ={v∈V |∂v=λ(∂)vfor all∂∈ L0}.

The spaceVλ is called theweight space corresponding to the weightλ. Thesupport of the weight moduleV, denoted by Supp(V), is the set of all weightsλwithVλ=0.

A weight moduleV is aHarish-Chandra module if dimV_λ<∞for allλ∈Supp(V) and is calleduniformly bounded if there is someN∈Nsuch that dimVλ≤N for all λ∈Supp(V).

Letgl_dbe the Lie algebra of alld×dcomplex matrices andsldbe the subalgebra of gl_d consisting of all traceless matrices. For 1≤i, j≤d we use Eij to denote the matrix units, i.e.,E_i,j has entry 1 at (i, j) and 0 otherwise.

For any integral dominant weightψ(on the Cartan subalgebra ofsld consisting of all diagonal matrices), letV(ψ) be the irreducible finite-dimensionalsld-module with highest weight ψ. We make V(ψ) into a gl_d module by defining the action of the identity matrixI as some scalar e∈C. We denote the resulting module by V(ψ, e).

For anyα∈C^d, it is known thatF^α(ψ, e)=V(ψ, e)⊗Ais aW-module by defin- ing

D(γ, n)(v⊗t^m) = (γ|m+α)v⊗t^m+n+((nγ^T)v)⊗t^m+n

for any m, n∈Z^d, γ∈C^d and v∈V(ψ, e). Moreover, given any c∈C, we can make F^α(ψ, e) into a (WA)-module by defining

tⁿ(v⊗t^m) =cv⊗t^m+n

for any m, n∈Z^d and v∈V(ψ, e). We will denote the resulted (WA)-module by F^α(ψ, e, c). It is obvious thatF^α(ψ, e) andF^α(ψ, e, c) are exp-polynomial modules in the sense of [4]. We now collect some known results on the modulesF^α(ψ, e) and F^α(ψ, e, c) from [5], [6] and [10].

Theorem 2.1. (1)F^α(ψ, e)is reducible as aW-module if and only if (ψ, e)=

(δk, k); or ψ=0, e∈{0, d} and α∈Z^d, where δk is the k-th fundamental weight of sld.

(2)F^α(ψ, e, c)is irreducible as a (WA)-module if c=0.

(3)Let V be an irreducible Harish-Chandra (WA)-module satisfyingt⁰v=v and t^mtⁿv=t^m+nv for any m, n∈Z^d and v∈V, then V∼=F^α(ψ, e,1) for suitable α∈C^d, e∈C, and someψ which is an integral dominant weight of sld.

2.2. Heisenberg–Virasoro algebra

We first recall the definition of the Heisenberg–Virasoro algebra HVir overC. Definition2.2. Thetwisted Heisenberg–Virasoro algebra HVir is a Lie algebra overCwith the basis

{xi, I(i), CD, CDI, CI|i∈Z}

and the Lie bracket given by

[xi, xj] = (j−i)xi+j+δi,−j

i³−i 12 CD, [x_i, I(j)] =jI(i+j)+δ_i,−j(i²+i)C_DI, [I(i), I(j)] =iδi,−jCI,

[HVir, CD] = [HVir, CDI] = [HVir, CI] = 0.

We can define weight modules over HVir by requiring that the action of the Cartan subalgebra Cx0⊕CI(0)⊕CCD⊕CCDI⊕CCI is diagonalizable. Then we have the concepts of Harish-Chandra modules and uniformly bounded modules similarly as we did for W and WA. In particular, for anya, b, c∈Cwe have the module of intermediate series V(a, b, c) over HVir which has a C-basis {vi|i∈Z}

and the HVir-actions

xjvi= (a+i+bj)vi+j, I(j)vi=cvi+j,

CDvi=CIvi=CDIvi= 0 fori, j∈Z.

And V(a, b, c) is reducible if and only if a∈Z, b∈{0,1} and c=0. The unique nontrivial irreducible subquotient ofV(a, b, c) is denoted byV(a, b, c).

In [17], Lu and Zhao gave a complete classification of irreducible Harish- Chandra modules over HVir. In particular, we have the following theorem.

Theorem 2.3. Any nontrivial irreducible uniformly bounded module over HVir is isomorphic toV(a, b, c)for somea, b, c∈C.

We note that for any fixed γ∈C^d and m∈Z^d with (γ|m)=0, the subalge- bra ofWAspanned by {D(γ, im), t^im|i∈Z}, which we denote by HVir(γ, m), is isomorphic to HVir/(CC_D⊕CC_DI⊕CC_I) by assigning D(γ, im)/(γ|m) to x_i and t^im/(γ|m) to I(i).

3. Uniformly bounded modules over WA

In this section, we give a description of irreducible uniformly bounded modules overWA with nontrivial action ofA.

We first recall a result on Z^d-graded A-modules. Here and later, we consider Aas aZ^d-graded commutative Lie algebra. LetV be aZ^d-graded module overA, that is,V=

m∈Z^dVmandtⁿVm⊆Vm+n for anym, n∈Z^d. AZ^d-gradedA-module is said to beirreducibleif it does not contain any nonzero proper graded submodule.

Lemma 3.1. LetV=

n∈Z^dVn be an irreducible uniformly boundedZ^d-graded A-module,then dimV_n≤1 for all n∈Z^d.

Proof. For n∈Z^d, let Un be the nth homogeneous subspace of U(A) with respect to the Z^d-gradation on A. Then U₀ is a commutative associative alge- bra, and each nonzero homogeneous subspaceV_n is a finite-dimensional irreducible

U0-module. The finite-dimensional condition guarantees that the action of U0 on Vn has common eigenvectors. Therefore dimVn≤1 for all n∈Z^d.

We will frequently use the following technical lemma later in this paper.

Lemma 3.2. Suppose that V is an irreducible (WA)-module and g∈U(A).

If gv=0for some nonzero v∈V, theng is locally nilpotent on V. Proof. For any γ∈C^d ands∈Z^d, we have

[[D(γ, s), g], g] = 0.

Therefore, we can show that g^l+1D(γ1, s1)...D(γl, sl)v

=

i1+...+il=l+1

(l+1)!

i1!...il!((adg)ⁱ¹D(γ1, s1))...((adg)^ilD(γl, sl))v

= 0,

where γi∈C^d and si∈Z^d, 1≤i≤l. Since V is irreducible, we see that g is locally nilpotent onV.

Theorem 3.3. Let V be an irreducible uniformly bounded weight (WA)- module. Let t⁰v=cvfor any v∈V,wherec∈C. Then the following are true:

(1) If c=0, then (t^mt^s−ct^m+s)v=0 for any m, s∈Z^d and v∈V. In this case, V∼=F^α(ψ, e, c)for suitableα∈C^d,e∈C,andψwhich is an integral dominant weight of sld.

(2) If c=0, thenAV=0.

Proof. There exists α∈C^d such thatV=

m∈Z^dVm, where

Vm={v∈V |D(δ,0)v= (δ|α+m)v for anyδ∈C^d} form∈Z^d.

(1) Suppose c=0. We may assume that c=1 by replacing t^m with t^m/c for anym∈Z^dif necessary, which is equivalent to making a Lie algebra automorphism.

Chooseγ∈C^d such that (γ|m)=0 for all nonzero m∈Z^d.

Claim 1. t^m acts injectively and t^mt^−m−t⁰ acts locally nilpotently on V for any m∈Z^d.

For any 0=m∈Z^d, the subalgebra HVir(γ, m) =

i∈Z

CD(γ, im)

i∈Z

Ct^im

is isomorphic to the Heisenberg–Virasoro algebra moduloCCD⊕CCDI⊕CCI and V(m)=

i∈ZVimis a uniformly bounded HVir(γ, m)-module. There is a nontrivial irreducible HVir(γ, m)-submoduleX ofV(m). Then (t^imt^jm−t^(i+j)m)v=0 for any i, j∈Z+ and v∈X, which implies by Lemma 3.2 that t^imt^jm−t^(i+j)m is locally nilpotent onV. Claim1 is proved.

Claim 2. For anym, n∈Z^d,there existsλm,n∈C^∗ such thatt^mtⁿ−λm,nt^m+n acts locally nilpotently on V.

Since V is a uniformly bounded Z^d-graded module over the Lie algebra A, by Lemma3.1, we can see that V contains an irreducible Z^d-gradedA-submodule X=

n∈Z^dX_n such thatX_n⊆V_n and dimX_n≤1. If there is somen∈Z^d such that Xn=0, then we can take any 0=v∈Xm for somem∈Z^d to deduce a contradiction to Claim1, viz. t^n−mv=0.

Thus we have dimXm=1 for allm∈Z^d and there existλm,n∈C^∗ satisfying t^mtⁿv=λ_m,nt^m+n form, n∈Z^d andv∈X.

Also by Lemma 3.2, t^mtⁿ−λ_m,nt^m+n acts locally nilpotently on V. Claim 2 is proved.

Claim 3. λ_m,n=1for all m, n∈Z^d.

Note that the claim is true forn=−mby Claim1. Set Xm,n=t^mtⁿ−λm,nt^m+n.

Note that dimVm=dimVn for allm, n∈Z^d. Choose a basis{v1, ..., vk} ofV0. Then {t^mv₁, ..., t^mv_k} is a basis of V_m by Claim1, and there existsB_m,n∈M_k(C) such that

t^mtⁿ(v1, ..., vk) =t^m+n(v1, ..., vk)Bm,n. Fromt^rt^st^mtⁿ=t^mtⁿt^rt^s,we see that

Br,sBm,n=Bm,nBr,s.

By Lie’s theorem, there exists a matrixS∈Mk(C) such thatCm,n=S⁻¹Bm,nS are upper triangular matrices for allm, n∈Z^d. Let

(w1, ..., wk) = (v1, ..., vk)S.

Then

(3) t^mtⁿ(w₁, ..., w_k) =t^m+n(w₁, ..., w_k)C_m,n. Consequently

(4) Xm,n(w1, ..., wk) =t^m+n(w1, ..., wk)(Cm,n−λm,nIk),

whereI_k is the identity matrix inM_k(C). By the fact thatX_m,nis locally nilpotent onV, we see that all the diagonal entries ofCm,nareλm,n. We obtain that (5) X_m,n^k V₀= 0 and X_m,nw₁= 0.

We can compute

0 =D(γ,−m−n)^kX_m,n^k (w₁, ..., w_k)

= (k![D(γ,−m−n), Xm,n]^k+Xm,ng₋m−n)(w1, ..., wk), (6)

where g_−m−n∈U(WA) is some element such thatg_−m−nV₀⊆V_−m−n. From (3) we have that

[D(γ,−m−n), X_m,n]^k(w₁, ..., w_k)

= ((γ|m)t⁻ⁿtⁿ+(γ|n)t^−mt^m−(γ|m+n)λm,nt⁰)^k(w1, ..., wk)

= (w1, ..., wk)((γ|m)C₋n,n+(γ|n)C₋m,m−(γ|m+n)λm,nIk)^k (7)

and from (3) and (4) that

Xm,ng₋m−n(w1, ..., wk) =t^−m−nXm,n(w1, ..., wk)G₋m−n

= (w₁, ..., w_k)(C_m+n,−m−n(C_m,n−λ_m,nI_k)G_−m−n), (8)

whereG₋m−n∈Mk(C) is given by

g₋m−n(w1, ..., wk) =t^−m−n(w1, ..., wk)G₋m−n. Note that C_m,nC_r,s=C_r,sC_m,n. Then thekth row of the matrix

Cm+n,−m−n(Cm,n−λm,nIk)G₋m−n= (Cm,n−λm,nIk)Cm+n,−m−nG₋m−n

is zero and all the diagonal entries of the upper triangular matrixC₋n,n are equal to 1 for alln∈Z^d. Combining (5), (6) and (8) and considering the coefficient ofwk

in D(γ,−m−n)^kX_m,n^k w_k, we have

(γ|m+n)^k(1−λm,n)^k= 0, that is,λm,n=1 for allm, n∈Z^d. Claim3is proved.

As a result of the formula (5) and Claim 3, we have t^mtⁿw1=t^m+nw1 for all m, n∈Z^d.So the set

V={v∈V |t^mtⁿv=t^m+nvfor allm, n∈Z^d}

is nonzero. It is easy to check that span{t^mtⁿ−t^m+n|m, n∈Z^d} is stable under the action of W, so V is a (WA)-submodule of V. By the irreducibility of the (WA)-module V, we see that V=V. Our result for this case follows from Theorem2.1.

(2) Suppose now that c=0. Since V is uniformly bounded, there exists N∈N such that dimVm<N for all m∈Z^d. For n∈Z^d, view

i∈ZVn+im as a module over HVir(γ, m). The fact that

i∈ZV_n+im is uniformly bounded implies that

i∈ZV_n+im has an HVir(γ, m)-modules composition series with the number of nontrivial composition factors not bigger thanN. By Theorem2.3,t^macts trivially on each composition factor. We can also see that the total number of composition factors (trivial and nontrivial) in this composition series is not bigger than 2N. So (t^m)^2NV=0 for all m∈Z^d. Let N0 be the smallest integer such that (t^m)^N0V=0 for allm∈Z^d. Clearly,N0≥1. Recall that (γ|r)=0 for allr∈Z^d.Then

0 =D(γ, r−m)(t^m)^N0V = (γ|m)N₀t^r(t^m)^N0−1V for allr, m∈Z^d.

By the choice ofN0, there are somev∈V and 0=m∈Z^dsuch thatu=(t^m)^N0−1v=0.

Thus,t^ru=0 for allr∈Z^d.

It is easy to check that the nonzero subspace

V={u∈V |t^ru= 0 for all r∈Z^d}

is a (WA)-submodule ofV. By the irreducibility ofV, we haveV=V. Part (2) follows.

4. Unbounded weight modules overWA

To describe irreducible weight modules over WA which are not uniformly bounded, we need to introduce some new notation. Let L=

m∈Z^dLm be a Z^d- graded Lie algebra such that L0 is abelian and the gradation itself is the root space decomposition with respect toL0. We also assume that [Lm,Ln]=Lm+n for all distinct m, n∈Z^d. It is clear that W, WA and (centerless) higher-rank or classical Virasoro algebras are examples of such algebras. We can define weight modules and Harish-Chandra modules and other similar concepts forL as we did for the algebrasW andWA.

For convenience, we denote by λn the weight of Ln in the adjoint represen- tation of L, that is, Ln={y∈L|[x, y]=λn(x)y for allx∈L0}, where n∈Z^d. When considering weights with respect toL, we identify{λn|n∈Z^d}with Z^d naturally.

Suppose thatV is a weightL-module. ThenV isdenseprovided Supp(V)=λ+

Z^dfor someλ∈L^∗0, and iscutprovided there areλ∈Supp(V),δ∈R^d\{0}andm∈Z^d such that Supp(V)⊆λ+m+Z^(δ)_≤0, whereZ^(δ)_≤0={n∈Z^d|(δ|n)≤0}. An elementv∈V is called ageneralized highest weight vectorprovided there exist aZ-basis{s1, ..., sd} ofZ^dandN∈Nsuch thatLmv=0 for allm=d

i=1m_is_i∈Z^dwithm_i>N,i=1, ..., d.

Let us recall the following general result forL-modules, see Theorem 4.1 in [19].

Theorem 4.1. LetV be an irreducible weightL-module,which is neither dense nor trivial. IfV contains a generalized highest weight vector,thenV is a cut module.

Now we introduce a class of cut modules overL. Choose some subgroupH⊂Z^d such thatZ^d=H⊕Zβ for someβ∈Z^d\{0}. Define the following subalgebras ofL:

LH=

g∈H

Lg, L⁺H=

g∈H k∈N

Lg+kβ and L⁻H=

g∈H k∈N

Lg−kβ.

LetX be an irreducible LH-module. We makeX into an (LH+L⁺H)-module by defining L⁺_HX=0. Then we define thegeneralized Verma module M(H, β, X) overL as

M(H, β, X) = Ind^L_L+

H+LHX=U(L)⊗_LH+L⁺

HX.

It is easy to see that M(H, β, X) has a unique irreducible quotient module, which we denote byL^L(H, β, X). If there are no ambiguities occurring, we simply denote it byL(H, β, X).

We will simply callL(H, β, X) anirreducible highest weight module. It is easy to see that L(H, β, X) is a cut module. It was shown in [4] that L(H, β, X) is a Harish-Chandra module ifX is a uniformly bounded exp-polynomial module.

Let γ∈C^d be such that (γ|m)=0 for all m∈Z^d. The subalgebra Vir(γ)=

m∈Z^dCD(γ, m) ofWAis a centerless rank-dVirasoro algebra (in the sense of [20]). We can see that L^Vir(γ)(H, β, X) is not uniformly bounded provided that X is nontrivial, by [16]. On the other hand, ifX is trivial, so isL^Vir(γ)(H, β, X).

Similar results hold forWAby considering (WA)-modules as Vir(γ)-modules.

Theorem 4.2. Any irreducible Harish-Chandra Vir(γ)-module which is not uniformly bounded is isomorphic to L^Vir(γ)(H, β, X) for some H, β andX, where β∈Z^d\{0}, H is a subgroup of Z^d such that Z^d=H⊕Zβ, and X is a nontrivial irreducible uniformly bounded Vir(γ)_H-module.

Using Theorems 4.1 and4.2, we can give a description of irreducible Harish- Chandra (WA)-modules which are not uniformly bounded.

Theorem 4.3. LetV be an irreducible Harish-Chandra(WA)-module which is not uniformly bounded. Then V is isomorphic to some L(H, β, X), where β∈ Z^d\{0}, H is a subgroup of Z^d such that Z^d=H⊕Zβ, and X is an irreducible uniformly bounded (WA)H-module.

Proof. For convenience setL=WA. By the irreducibility ofV, there is some λ∈L^∗0 such that Supp(V)⊆λ+Z^d. Let {ei|i=1, ..., d} be the canonical Z-basis of Z^d. Given anyn∈Z^d, we will denote by ni theith entry ofnfori=1, ..., d, that is, n=d

i=1n_ie_i.

Choose anyγ∈C^d such that (γ|m)=0 for allm∈Z^d. ThenV can be viewed as a Vir(γ)=

n∈Z^dCD(γ, n)-module, which is not uniformly bounded. Moreover each V(i)=

n∈Z^d,n1=0Vλ+ie₁+nfori∈Zis a module over

n∈Z^d,n1=0CD(γ, n), which is a centerless rank-(d−1) Virasoro algebra for anyi∈Z, andV=

n∈Z^d,n2=0V_λ+n is a module over

n∈Z,n₂=0CD(γ, n), which is also a centerless rank-(d−1) Virasoro algebra.

Note that any uniformly bounded module over classical or higher-rank Virasoro algebras has equal dimensions of weight spaces except for the weight 0. If allV(i), i∈Z, andV are uniformly bounded, then there existsN∈Nsuch that dimVλ+n≤ N for alln∈Z^d withn2=0. SetN=max{N, δ_λ,ZddimV0}, whereδ_λ,Zd=1 ifλ∈Z^d andδ_λ,Zd=0 otherwise. Since V(i) is uniformly bounded, we have

dimVλ+ie₁+m≤max{dimVλ+ie₁+m−m₂e₂, δ_λ,ZddimV0} ≤N

for all i∈Z and m∈Z^d with m1=0. In other words, dimVλ+m≤N for all m∈Z^d, contradicting the fact thatV is not uniformly bounded. Thus, eitherV(i), for some i∈Z, orV is not uniformly bounded.

Without loss of generality, we may assume thatV(0) is not uniformly bounded.

So there exists somen∈Z^d withn1=0 such that (9) dimVλ−n>(d+1)

dimVλ+e₁+

d

i=2

dimVλ+e₁+e_i

.

We sets₁=n+e₁ands_i=n+e₁+e_ifori=2, ..., d. It is easy to check that the set{s_i| i=1, ..., d}is aZ-basis of Z^dsincen=d

i=2n_ie_i. By (9), there exists some nonzero elementv∈Vλ−nsuch thatD(ei, sj)v=0 andt^sjv=0 for anyi, j=1, ..., d. Thus,vis a generalized highest weight vector relative to the basis{si|i=1, ..., d}ofZ^d. By the irreducibility of the L-moduleV and the Poincar´e–Birkhoff–Witt theorem, we see thatd

j=1Nsj∈/Supp(V). Clearly,V is neither dense nor trivial. By Theorem4.1,

there exists somem∈Z^d andδ∈R^d\{0}such that Supp(V)⊆λ+m+Z^(δ)_≤0. SetG_δ= {n∈Z^d|(δ, n)=0}.

Now consider V as a Vir(γ)-module. Then V has an irreducible nontrivial Vir(γ)-subquotient, say Y, which is not uniformly bounded. By the representa- tion of higher-rank Virasoro algebras, we know thatY∼=L^Vir(γ)(H, β, Y) for some β∈Z^d\{0}, a subgroupH ofZ^d withZ^d∼=H⊕Zβ, andYbeing an irreducible uni- formly bounded module overLH∩Vir(γ) which is a rank-(d−1) centerless Virasoro algebra. In particular,

λ−iβ+H⊆Supp(Y)⊆Supp(V)⊆λ+m+Z^(δ)_≤0 for sufficiently largei∈N. It follows that

(δ|n)≤(δ|m+iβ) forn∈H and sufficiently largei∈N, yielding that H=G_δ and (δ|β)>0.

Leti₀∈Zbe the maximal number such thatX=V_λ+i0β+H=0. Thus, Supp(V)⊆

i≤i₀(λ+iβ+H) and L⁺HX=0. Note that X is an LH-module. Then the irre- ducibility ofV overLimplies the irreducibility ofXoverLHand thatV=U(L⁻H)X. SinceV is nontrivial overL, we know thatXis nontrivial overLH. IfXis not a uniformly boundedLH-module, thenX has an irreducibleLH∩Vir(γ)-subquotient Xwhich is not uniformly bounded. The unique irreducible Vir(γ)-quotient module of the induced Vir(γ)-module

U(Vir(γ))⊗_(LH+L⁺

H)∩Vir(γ)X

is not a Harish-Chandra module by Theorem 2.3(2) of [16], contradicting the fact that V is a Harish-Chandra L-module. Thus we must have that X is uniformly bounded overLH. This completes the proof.

Using Theorems3.3and4.3, we can obtain a classification of irreducible Harish- Chandra modules overWAwith nontrivial action ofA.

Theorem 4.4. Suppose that V is a nontrivial irreducible Harish-Chandra (WA)-module with nonzero action of A.

(1) If V is uniformly bounded, then V is isomorphic to some F^α(ψ, e, c) for suitableα∈C^d, e, c∈C,andψ which is an integral dominant weight of sld.

(2) If V is not uniformly bounded, then V is isomorphic to some L(H, β, X), where β∈Z^d\{0}, H is a subgroup of Z^d such that Z^d=H⊕Zβ, and X is a non- trivial irreducible uniformly bounded (WA)H-module.

Acknowledgements. We are grateful to the referee for giving many good sug- gestions to make the paper more readable. Xiangqian Guo was partially supported by the NSF of China (Grant No. 11101380). Kaiming Zhao was partially supported by NSERC.

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Xiangqian Guo

Department of Mathematics Zhengzhou University Zhengzhou, Henan 450001 P.R. China

[email protected] Genqiang Liu

College of Mathematics and Information Science

Henan University Kaifeng, Henan 475004 P.R. China

[email protected]

Kaiming Zhao

Department of Mathematics Wilfrid Laurier University Waterloo, ON N2L 3C5 Canada

[email protected] and

College of Mathematics and Information Science

Hebei Normal (Teachers) University Shijiazhuang, Hebei 050016

P.R. China

Received May 10, 2012 in revised form June 23, 2012 published online September 15, 2012